# Hypergeometric differential equation

In

mathematics , the**hypergeometric differential equation**is a second-order linearordinary differential equation (ODE) whose solutions are given by the classical hypergeometric series. Every second-order linear ODE with threeregular singular point s can be transformed into this equation. The solutions are a special case of aSchwarz-Christoffel mapping to a triangle with circular arcs as edges. These are important because of the role they play in the theory oftriangle group s, from which the inverse to Klein'sJ-invariant may be constructed. Thus, the solutions are coupled to the theory ofFuchsian group s and thus hyperbolicRiemann surface s.**Definition**The hypergeometric differential equation is

:$z(1-z)frac\; \{d^2w\}\{dz^2\}\; +\; left\; [c-(a+b+1)z\; ight]\; frac\; \{dw\}\{dz\}\; -\; abw\; =\; 0.$

It has three regular singular points: 0,1 and ∞. The generalization of this equation to arbitrary regular singular points is given by

Riemann's differential equation .**olutions**Solutions to the differential equation are built out of the

hypergeometric series $;\_2F\_1(a,b;c;z).$ In general, the equation has twolinearly independent solutions. One starts by defining the values:$lambda=1-c$:$mu=c-a-b$:$u=a-b.$

These are known as the

**angular parameters**for the regular singular points 0,1 and ∞ respectively. Frequently, the notation $u\_0$, $u\_1$ and $u\_infty$, respectively, are used for the angular parameters. Sometimes, the**exponents**$mu\_0$, $mu\_1$, $mu\_z$ and $mu\_infty$ are used, with:$mu\_0=frac\{1\}\{2\}(1-\; u\_0+\; u\_1-\; u\_infty)=\; c-a$:$mu\_1=frac\{1\}\{2\}(1+\; u\_0-\; u\_1-\; u\_infty)=\; b+1-c$:$mu\_z=frac\{1\}\{2\}(1-\; u\_0-\; u\_1+\; u\_infty)=\; a$:$mu\_infty=frac\{1\}\{2\}(1+\; u\_0+\; u\_1+\; u\_infty)=\; 1-b$

and $mu\_0+mu\_1+mu\_z+mu\_infty=2$.

The general case, where none of the angular parameters are

integer s, is given below. When one or more of these parameters are integers, the solutions are given in the articlehypergeometric equation solutions .Around the point "z"=0, the two independent solutions are

:$phi\_0^\{(0)\}(z)=\; ;\_2F\_1(a,b;c;z)$

and

:$phi\_0^\{(1)\}(z)\; =\; z^lambda\; ;\_2F\_1(a+lambda,b+lambda;1+lambda;z)$

Around "z"=1, one has

:$phi\_1^\{(0)\}(z)=\; ;\_2F\_1(a,b;1-mu;1-z)$

and

:$phi\_1^\{(1)\}(z)\; =\; (1-z)^mu\; ;\_2F\_1(b+mu,a+mu;1+mu;1-z)$

Around "z"=∞ one has

:$phi\_infty^\{(0)\}(z)\; =\; z^\{-a\};\_2F\_1(a,a+lambda;1+\; u;\; z^\{-1\})$

and

:$phi\_infty^\{(1)\}(z)\; =\; z^\{-b\};\_2F\_1(b,b+lambda;1-\; u;\; z^\{-1\})$

This is the complete set of solutions. Kummer's set of 24 canonical solutions may be obtained by applying either or both of the following identities to the above equations:

:$;\_2F\_1(a,b;c;z)=\; (1-z)^\{c-a-b\}\; ;\_2F\_1(c-a,c-b;c;z)$

and

:$;\_2F\_1(a,b;c;z)=(1-z)^\{-a\}\; ;\_2F\_1(a,c-b;c;z/(z-1))$

For a solution of this differential equation using Frobenius method, please check

Frobenius solution to the hypergeometric equation .**Connection coefficients**Pairs of solutions are related to each other through connection coefficients, corresponding to the analytic continuation of the solutions. Denote a pair of solutions as the column vector

:$Phi\_k\; =\; left(\; egin\{matrix\}\; phi\_k^\{(0)\}\; \backslash \; phi\_k^\{(1)\}end\{matrix\}\; ight)$

for "k"=0,1, ∞. Pairs are related by matrices

:$Phi\_0\; =\; left(\; egin\{matrix\}\; frac\{Gamma(c)Gamma(c-a-b)\}\{Gamma(c-a)Gamma(c-b)\}\; ;\; ;frac\{Gamma(c)Gamma(a+b-c)\}\{Gamma(a)Gamma(b)\}\; \backslash ;\; ;\; \backslash frac\{Gamma(2-c)Gamma(c-a-b)\}\{Gamma(1-a)Gamma(1-b)\}\; ;\; ;frac\{Gamma(2-c)Gamma(a+b-c)\}\{Gamma(a+1-c)Gamma(b+1-c)\}end\{matrix\}\; ight)Phi\_1$

and :$Phi\_0\; =\; left(\; egin\{matrix\}\; e^\{-ipi\; a\}\; frac\{Gamma(c)Gamma(b-a)\}\{Gamma(c-a)Gamma(b)\}\; ;\; ;e^\{-ipi\; b\}\; frac\{Gamma(c)Gamma(a-b)\}\{Gamma(c-b)Gamma(a)\}\; \backslash ;\; ;\; \backslash e^\{-ipi(a+1-c)\}\; frac\{Gamma(2-c)Gamma(b-a)\}\{Gamma(b+1-c)Gamma(1-a)\}\; ;\; ;e^\{-ipi(b+1-c)\}\; frac\{Gamma(2-c)Gamma(a-b)\}\{Gamma(a+1-c)Gamma(1-b)\}end\{matrix\}\; ight)Phi\_infty$

where Γ is the

gamma function .**Q-form**The hypergeometric equation may be brought into the Q-form

:$frac\{d^2u\}\{dz^2\}+Q(z)u(z)\; =\; 0$

by making the substitution $w=uv$ and eliminating the first-derivative term. One finds that

:$Q=frac\{z^2\; [1-(a-b)^2]\; +z\; [2c(a+b-1)-4ab]\; +c(2-c)\}\{4z^2(1-z)^2\}$

and "v" is given by the solution to

:$frac\{d\}\{dz\}log\; v(z)\; =\; frac\; \{c-z(a+b+1)\}\{2z(1-z)\}.$

The Q-form is significant in its relation to the

Schwarzian derivative .**chwarz triangle maps**The

**Schwarz triangle maps**or**Schwarz "s"-functions**are ratios of pairs of solutions.:$s\_k(z)\; =\; frac\{phi\_k^\{(1)\}(z)\}\{phi\_k^\{(0)\}(z)\}$

where "k" is one of the points 0, 1, ∞. The notation

:$D\_k(lambda,mu,\; u;z)=s\_k(z)$is also sometimes used. Note that the connection coefficients become

Möbius transformation s on the triangle maps.Note that each triangle map is

regular at "z" ∈ {0, 1, ∞} respectively, with:$s\_0(z)=z^lambda\; (1+mathcal\{O\}(z))$:$s\_1(z)=(1-z)^mu\; (1+mathcal\{O\}(1-z))$and:$s\_infty(z)=z^\; u\; (1+mathcal\{O\}(1/z)).$

In the special case of λ, μ and ν real, with $0le|lambda|,|mu|,|\; u|<1$ then the s-maps are

conformal map s of theupper half-plane **H**to triangles on theRiemann sphere , bounded by circular arcs. This mapping is a special case of aSchwarz-Christoffel map . The singular points 0,1 and ∞ are sent to the triangle vertices. The angles of the triangle are πλ, πμ and πν respectively.Furthermore, in the case of $lambda=1/p$, $mu=1/q$ and $u=1/r$ for integers "p", "q", "r", then the triangle tiles the sphere, and the s-maps are inverse functions of

automorphic function s for thetriangle group $langle\; p,q,r\; angle=Delta\; (p,q,r).$**Monodromy group**The monodromy of a hypergeometric equation describes how fundamental solutions change when analytically continued around paths in the "z" plane that return to the same point. That is, when the path winds around a singularity of $;\_2F\_1$, the value of the solutions at the endpoint will differ from the starting point.

Two fundamental solutions of the hypergeometric equation are related to each other by a linear transformation; thus the monodromy is a mapping (group homomorphism):

:$pi\_1(z\_0,mathbb\{C\}setminus\{0,1\})\; o\; GL(2,mathbb\{C\})$

where $pi\_1$ is the

fundamental group . In other words the monodromy is a two dimensional linear representation of the fundamental group.Themonodromy group of the equation is the image of this map, i.e. the group generated by the monodromy matrices.**ee also*** Second order ODE's with four regular singular points can always be transformed into

Heun's equation .

*Schottky group

*Schwarzian derivative

*Complex differential equation **References*** Milton Abramowitz and Irene A. Stegun, eds., "

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables" (Dover: New York, 1972)

** [*http://www.math.sfu.ca/~cbm/aands/page_556.htm Chapter 15*] Hypergeometric Functions

*** [*http://www.math.sfu.ca/~cbm/aands/page_562.htm Section 15.5*] The Hypergeometric Differential Equation* Frits Beukers, " [

*http://www.math.uu.nl/people/beukers/MRIcourse93.ps Gauss' hypergeometric function*] " (2002) "(Lecture notes reviewing basics, as well as triangle maps and monodromy)"* cite book|author=Masaaki Yoshida

title=Hypergeometric Functions, My Love: Modular Interpretations of Configuration Spaces

year=1997

publisher=Friedrick Vieweg & Son

id=ISBN 3-528-06925-2

*Wikimedia Foundation.
2010.*

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